well i only did the Chamonix to arolla section back in 2013. I would like to do the other half one day. When are you going?
I did not like key reynolds book very much I don't know how many other options you have in English, so you'll probably end up getting it anyway. Maps are more helpful probably. My main tips are these:
1) bring cash on your way - you can stop and get more at several towns. you can get demi-pension and then order packed lunches at the huts - so you needn't take ANY food at all with you. It's fully catered up there. That's the swiss way.
2) day one is only a half day walk and so you could start in the arvo and still reach the hut in time for din dins.
3) day 2 or 3(???) can't remember - there is an option to take the "fenetre" or go around it. Answer: take the fenetre. Longer but this is one of the big scenery days.
4) day 3 or 4 ?? there is a SUPER long day with 1800mtrs ascent. You will be in sun much of the day and may have trouble finding the route. I think there was some public transport option for part of that day and tbh, it is not a very scenic day - so investigate that option. it might have been between champex and cabane du mont fort.
5) i wish i had taken the extra day to stop at a hut that goes up to a glacier on the way between the cabane du mont fort and cabane de louvie huts. don't know the hut's name sorry.
6) i really liked the atmosphere at the cabane de louvie - but that was due to staff and the same people may not be there this year.
7) and that brings up to my last point OMG JUST PREPARE YOURSELF for miserable swiss hut wardens - even when you hand over your hard earned they will all have a scowl on their face (except at louvie), don't take it to heart - if they can't be happy living in paradise, they can't be happy anywhere.
oh wait point
i think if you join the britannica section of the austrian alpine club you will get reduced fees each night and then also mountain rescue insurance - the thing will pay for itself if you only need to save on hut fees. see here: http://aacuk.org.uk